Relate the terhmal coefficient of linear expansion, `alpha`, the the thermal coefficient of area expansion `beta`.

Explanation

A solid may have different coefficients of linear expansion for different directions but for simplicity we can assume that the same coefficient applies to all directions, i.e., the solids show isotropic expansion.
Since area, A is length squared, `L^(2)`
`A=L^(2)=L_(0)^(2)(1+alphaDeltaT)^(2)=A_(0)(1+2alphaDeltaT+alpha^(2)DeltaT^(2))`.
Since `alpha lt lt 1` (in fact, `alphacong10^(-6)` for solids), the second-order term (containing `alpha^(2)`) can be dropped with negligible error. we then obtain
`A=A_(0)(1+2alphaDeltaT)` or `(DeltaA)/(A_(0))~~2alphaDeltaT`
Thus, the terhmal coefficient of area expansion, `beta` is twice as large as the coefficient of linear expansion, i.e., `bet~~2alpha`.